# GDS Eingangstest Zahlensysteme

## Versuch 3

### 1

$$ (27)_b + (28)_b = (54)_b $$

$$ (2b + 7 + 2b + 8)_{10} = (5b + 4)_{10} $$
$$ 4b + 15             = 5b + 4 $$
$$      11             = b $$

### 2

$$ A=(1101)_2 $$
$$ B=(10100)_2 $$

```
 (10100)_2 * (1101)_2 = (100000100)_2
  10100000
 + 1010000
 +   10100
 100000100
```

$$ (10100)_2 * (1101)_2 = (100000100)_2 $$

### 3

$$ zk ... zweierkomplement $$

$$ v_{zk} = (001000100)_2 $$
$$ (v_{zk})_{10} = 2^2 + 2^6 = 4 + 64 = 68 $$

### 4

$$ ez ... exzess $$
$$ z_{ez} = (1100111)_2 $$

$$ z = -(8)_{10} $$
$$ (z_{ez})_{10} = 2^0 + 2^1 + 2^2 + 2^5 + 2^6 $$
          $$ = 1 + 2 + 4 + 32 + 64 $$
          $$ = 103 $$

$$ e = (103 - (-8))_{10} = (111)_{10} $$

$$ 111 mod 2 = 1 ; 110 / 2 = 55 $$
$$ 55  mod 2 = 1 ;  54 / 2 = 27 $$
$$ 27  mod 2 = 1 ;  26 / 2 = 13 $$
$$ 13  mod 2 = 1 ;  12 / 2 = 6 $$
$$ 6   mod 2 = 0 ;   6 / 2 = 3 $$
$$ 3   mod 2 = 1 ;   2 / 2 = 1 $$
$$ 1   mod 2 = 1 ;   0 / 2 = 0 $$

$$ (e)_2 = (1101111)_2 $$

### 5

$$ (10000.11001)_2 = (x)_{10} $$

$$ (x)_{10} = 2^4 + 2^{-1} + 2^{-2} + 2^{-5} $$
$$          =  16 +  1/2 +  1/4 + 1/32 $$
$$          =  16 +  0.5 + 0.25 + 0.031 $$
$$          = 16.781 $$

```
1/32=0_031
100
  40
   8R
```

```
 0_5
+0_25
+0_031
=0_781
```

### 6

$$ (455.592)_{10} = (x)_{16} $$

```
455/16 = 28
135
  7R
```

$$ (455-7)/16 = 448/16 = 28 $$

```
28/16 = 1
12R
```

$$ (28-12)/16 = 16/16 = 1 $$

```
1/16 = 0
1R
```

$$ (x)_{16} = (1C7)_{16} + (0.592)_{10} $$

```
0.592 * 16 = 9.472
5.920
   12
  540
3.000
```

```
0.472 * 16 = 7.552
4.720
   12
  420
2.400
```

```
0.552 * 16 = 8.832
5.520
   12
  300
3.000
```

$$ (x)_{16} = (1C7.978)_{16} $$

### 7

$$ (DB4.E07)_{16} = (x)_2 $$

```
4 = 0100
7 = 0111
B = 1011
C = 1100
D = 1101
E = 1110
```

$$ (x)_2 = (110110110100.111...)_2 $$

### 8

$$ (356.077)_8 = (x)_{10} $$

$$ (x)_{10} = 3*8^2 + 5*8^1 + 6*8^0 + 0*8^-1 + 7*8^-2 + 7*8^-3 $$
$$          =  3*64 +   5*8 +     6 +      0 +   7/64 +  7/512 $$
$$          =   192 +    40 +     6 +           56/512 + 7/512 $$

$$ 3*512 = 1024 + 512 = 1536 < 1560 $$

```
63/512 = 0.123...
630
1180
 1560
  ...R
```

```
 192
+ 40
+  6
=238
```

$$ (x)_{10} = (238.123...)_{10} $$

### 9

$$ (0.8125)_{10} = (0.11...)_2 $$

```
0.8125 * 2 = 1.625
    10
    40
   200
1.6000
```

```
0.625 * 2 = 1.25
   10
   40
1.200
```

$$ (0.11)_2 = 1/2 + 1/4 = (0.75)_{10} $$

$$ (0.8125)_{10} - (0.75)_{10} = 0.0625 $$
